Incomplete vertices in the prime graph on conjugacy class sizes of finite groups.
This paper deals with the prime graph \(\Pi(G)\) associated to the conjugacy classes of a finite group \(G\). Recall that the vertices of \(\Pi(G)\) are the prime numbers dividing some conjugacy class size of \(G\) and two distinct primes are joined by an edge when there is some class size divisible by both. The authors study how the set of incomplete vertices (i.e., vertices which are not joined to all the vertices) exerts a strong influence on the structure of the group.NEWLINENEWLINE The main result is a generalization of a former theorem by the same authors [J. Algebra 364, 1-12 (2012; Zbl 1268.20033)], which proved that any finite group such that all the vertices of its prime graph are incomplete must be metabelian. In the paper under review, the authors show that if \(\pi\) denotes the set of incomplete vertices of the prime graph \(\Pi(G)\) of a group \(G\), then \(G\) possesses metabelian Hall \(\pi\)-subgroups, and if \(H\) is such a subgroup, then \(H\cap G''\leq\mathbf F(G)\). The proof relies on the classification of finite simple groups by means of a result due to the first and second author [Isr. J. Math. 174, 403-418 (2009; Zbl 1191.20024)].NEWLINENEWLINE Another theorem is the following. If \(G\) is a finite group, \(p\) is a vertex of \(\Pi(G)\) and \(\pi\) is a set of vertices of \(\Pi(G)\) that are non-adjacent to \(p\) and different from \(p\), then \(G\) is \(\pi\)-solvable, has Abelian Hall \(\pi\)-subgroups and \(\pi\)-length \(1\), and the vertices in \(\pi\) are pairwise adjacent.
scientific article; zbMATH DE number 6201455